Problem: Simplify and expand the following expression: $ \dfrac{-10}{4t - 3}+\dfrac{3t + 7}{3t - 2} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4t - 3)(3t - 2)$ Multiply the first term by $\dfrac{3t - 2}{3t - 2}$ $ \begin{align*} \dfrac{-10}{4t - 3} \times \dfrac{3t - 2}{3t - 2} & = \dfrac{(-10)(3t - 2)}{(4t - 3)(3t - 2)} \\ & = \dfrac{-30t + 20}{(4t - 3)(3t - 2)}\end{align*} $ Multiply the second term by $\dfrac{4t - 3}{4t - 3}$ $ \begin{align*} \dfrac{3t + 7}{3t - 2} \times \dfrac{4t - 3}{4t - 3} & = \dfrac{(3t + 7)(4t - 3)}{(3t - 2)(4t - 3)} \\ & = \dfrac{12t^2 + 19t - 21}{(3t - 2)(4t - 3)}\end{align*} $ Now we have: $ = \dfrac{-30t + 20}{(4t - 3)(3t - 2)} + \dfrac{12t^2 + 19t - 21}{(3t - 2)(4t - 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{-30t + 20 + 12t^2 + 19t - 21}{(4t - 3)(3t - 2)} $ $ = \dfrac{-11t - 1 + 12t^2}{(4t - 3)(3t - 2)}$ Expand the denominator: $ = \dfrac{-11t - 1 + 12t^2}{12t^2 - 17t + 6}$